Question

If alpha and beta are the zeroes of the quadratic polynomial : x²-7x+10, evaluate alpha³+beta³​

Answer 1

Answer :

α³ + ß³ = 133

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros .

★ A quadratic polynomial can have atmost two zeros .

★ The general form of a quadratic polynomial is given as ; ax² + bx + c .

★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;

• Sum of zeros , (α + ß) = -b/a

• Product of zeros , (αß) = c/a

Solution :

Here ,

The given quadratic polynomial is ;

x² - 7x + 10

Comparing the given quadratic polynomial with the general quadratic polynomial ax² + bx + c , we have ;

a = 1

b = -7

c = 10

Also ,

It is given that , α and ß are the zeros of the given quadratic polynomial .

Thus ,

=> Sum of zeros = -b/a

=> α + ß = -(-7)/1

=> α + ß = 7

Also ,

=> Product of zeros = c/a

=> αß = 10/1

=> αß = 10

Now ,

We know that ,

(a + b)³ = a³ + b³ + 3ab(a + b)

Thus ,

=> (α + ß)³ = α³ + ß³ + 3αß(α + ß)

=> 7³ = α³ + ß³ + 3×10×7

=> 343 = α³ + ß³ + 210

=> α³ + ß³ = 343 - 210

=> α³ + ß³ = 133

Hence , α³ + ß³ = 133 .